A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\,389\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\,389\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$

If $992\,466\,1A6$ is divisible by $8,$ where $A$ represents one digit, what is the sum of the possible values of $A?$
Solution: For $992\,466\,1A6$ to be divisible by $8,$ we must have $1A6$ divisible by $8.$ We check each of the possibilities, using a calculator or by checking by hand:

$\bullet$ $106$ is not divisible by $8,$ $116$ is not divisible by $8,$ $126$ is not divisible by $8,$

$\bullet$ $136$ is divisible by $8,$

$\bullet$ $146$ is not divisible by $8,$ $156$ is not divisible by $8,$ $166$ is not divisible by $8,$

$\bullet$ $176$ is divisible by $8,$

$\bullet$ $186$ is not divisible by $8,$ $196$ is not divisible by $8.$

Therefore, the possible values of $A$ are $3$ and $7.$ Thus, the answer is $7+3=\boxed{10}.$